The family of Jared Lee Loughner, the 22-year-old accused of shooting dead six people in Arizona in a rampage that left Democratic congresswoman Gabrielle Giffords critically injured, issued a statement yesterday expressing remorse over their son's actions.

"There are no words that can possibly express how we feel. We wish that there were so we could make you feel better. We don't understand why this happened. It may not make any difference, but we wish that we could change the heinous events of Saturday," the statement read.

"We care very deeply about the victims and their families. We are so very sorry for their loss."

Doctors treating Giffords for a bullet wound to the head yesterday gave her the most optimistic assessment yet, saying she could live to be 95. Dr Michael Lemole, the head of neurosurgery at the University Medical Centre in Tucson, said she was able to breathe on her own and was still responding to simple commands.

The funeral of one the victims, nine-year-old Christina Taylor Green, is to be held in Tucson, with a service on Friday for the judge who was killed in the attack, John Roll.

The Loughner family's statement came as security was stepped up for President Obama as he flew into Arizona for a memorial service for the victims of the weekend rampage.

Obama's address at the University of Arizona in Tucson comes against a backdrop of a nationwide debate about whether violent political rhetoric contributed to the shooting and, to a lesser extent, whether there is a need for new gun controls and increased security for members of Congress.

The US media has compared Obama's address to one Bill Clinton delivered in 1995 after the Oklahoma City bombing in which he criticised the rise of militias.

Obama has to tread carefully in a speech that will be carried live on television.

The shooting touched off a debate in the US between the left and right over whether violence-laden political imagery and rhetoric had contributed to the shooting.

Brian Miller, chairman of the local Republican party, told the website Talking Points Memo that Obama should "lead the effort to stop the blame".

Arizona offers special problems for the secret service in protecting Obama, given the state's lax gun control laws, which allow people to carry concealed weapons almost anywhere.

The memorial service, entitled Together We Thrive: Tucson and America, is open to the public.

In Washington, members of Congress are to discuss how to improve their personal security. Peter King, newly elected chairman of the house homeland security committee, is planning to introduce a bill to make it illegal to bring a gun within 1,000ft of a government official. It is an unusual step for a Republican, most of whom are supporters of gun rights.

King, who is from New York, is being supported by the New York mayor, Michael Bloomberg, a long-time advocate of gun controls.After the Virginia Tech massacre in 2007, there were similar calls for gun controls but legislation since then has gone the other way, with states making gun ownership easier.

Bloomberg said: "Civil debate is important, and I've long spoken out in favour of more co-operation and less antagonism, but I think it's true that the more we learn, the more it becomes clear that this case is fundamentally about a mentally ill drug abuser who had access to guns and shouldn't have."

The bill has little chance of making it into law, given the strength of the pro-gun lobby. The bill also appears difficult to implement in practice: how to establish whether there is government official in the vicinity.

The Republicans, who now control the House, had been planning to initiate a repeal of what they termed Obama's "job-killing" health care legislation. But, given the sensitivities round the issue, including Giffords' passionate support for healthcare reform, that has been postponed in favour of a discussion with the Capitol police and the FBI to discuss the safety of members of Congress.

Other measures include the introduction of toughened glass to provide a shield round the House public gallery, which is open at present, and a police presence at events in their home districts. There was no member of the police present at Giffords' public meeting on Saturday.

The security services have ruled out – on grounds of cost – providing protection round the clock for all 535 members of Congress, not least because there are only 1,800 police for the whole Capitol complex and they had their budget cut last year by 7%. Many members are opposed anyway to measures to restrict access to the public.

The Senate sergeant-at-arms said yesterday that senators he had spoken were resistant to being placed in a cocoon.

The Democratic member of Congress, Jesse Jackson junior, said he is to propose a reversal of a 5% cut Republicans made in members' office budgets and an extra 10% rise to cover more security.